Gamma and Generalized Gamma Distributions Victor Hugo Jiménez Nava University of Texas at El Paso, Vhjimeneznava@Miners.Utep.Edu

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2011-01-01 Gamma and Generalized Gamma Distributions Victor Hugo Jiménez Nava University of Texas at El Paso, [email protected]

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This is brought to you for free and open access by DigitalCommons@UTEP. It has been accepted for inclusion in Open Access Theses & Dissertations by an authorized administrator of DigitalCommons@UTEP. For more information, please contact [email protected]. GAMMA AND GENERALIZED GAMMA DISTRIBUTIONS

V´ICTOR HUGO JIMENEZ´ NAVA

Department of Mathematical Science

APPROVED:

Panagis Moschopoulos, Ph.D., Chair

Ori Rosen, Ph.D.

Naijun Sha, Ph.D.

Dr. Max Shpak, Ph.D

Benjam´ın C. Flores, Ph.D. Acting Dean of the Graduate School to my

MOTHER and FATHER

with love GAMMA AND GENERALIZED GAMMA DISTRIBUTIONS

V´ICTOR HUGO JIMENEZ´ NAVA

THESIS

Presented to the Faculty of the Graduate School of

The University of Texas at El Paso

in Partial Fulﬁllment

of the Requirements

for the Degree of

MASTER OF SCIENCE

Department of Mathematical Science

THE UNIVERSITY OF TEXAS AT EL PASO

December 2011 Acknowledgements

I would like to express my deep-felt gratitude to my advisor, Dr. Panagis Moschopoulos of the Mathematical Science Department at The University of Texas at El Paso, for his advice, encouragement, guidance and constant support. I believe he has been more than patient. He has always tried to lead me in the right direction and all this time working with him I have learned a lot. He has always tried of providing clear explanations when I was confused and lost. But he was there every time dealing with my impertinences when working and giving me his time, in spite of anything else that was going on. I really appreciate all he did. I do not find words to thank him for all his patience. I also wish to thank the other members of my committee, Dr. Naijun Sha, Dr. Ori Rosen, both of the Mathematical Science Department at The University of Texas at El Paso and Dr. Max Shpak of the Biological Science Department of the University of Texas at el Paso too. Their suggestions, comments and additional guidance were a big support to the completion of this work. I would like to thank as well to the Government of the State of Chihuahua Mexico for providing me with the financial support to study in this great university. They supported me with no complaints and always trying to make me a better citizen not only for the Chihuhua State but for all Mexico. I would like to thank as well to my parents for all his support in order to complete this master degree successfully. During all this time they have given to me their help, cooperation and motivating me to continue in this task of my life. I really thank them for all what they have done. Additionally, I want to thank The University of Texas at El Paso Mathematical Science Department professors and staff for all their hard work and dedication, providing me the means to complete my degree and prepare for a career as a statistician.

NOTE: This thesis was submitted to my Supervising Committee on the November 30, 2011.

iv Contents

Page Acknowledgements ...... iv Table of Contents ...... v List of Tables ...... vii List of Figures ...... x Chapter 1 Introduction ...... 1 2 The Gamma Distribution and its Basic Properties ...... 3 2.1 History and Motivation ...... 3 2.2 The Gamma Function ...... 3 2.3 Getting the Gamma Density ...... 5 2.4 Mean, Variance and Moment Generating Function ...... 5 2.5 Particular Cases and Shapes ...... 7 2.5.1 Standard Gamma ...... 7 2.5.2 The Exponential ...... 8 2.5.3 The Chi square distribution ...... 9 2.5.4 Linear Combinations of Independent Chi-Square ...... 10 3 Estimation of Parameters ...... 12 3.1 Method of Moments ...... 12 3.2 Maximum Likelihood Estimation ...... 13 3.3 Applications ...... 15 4 Distributions Related to the Gamma Distributions ...... 19 4.1 Log Gamma Distribution ...... 19 4.2 The Quotient of Two Independent Gamma ...... 20 X 4.3 The Quotient 1 ...... 20 X1 + X2

v 4.4 Sum of k Independent Gamma ...... 21 4.4.1 Aprroximation for the Exact Distribution of the sum of k Indepen- dent Gamma ...... 23 Approximation with Two Moments ...... 23 Normal Approximation ...... 23 5 Generalized Gamma Distributions ...... 37 5.1 History ...... 37 5.2 Motivation and Getting the Generalized Gamma Density ...... 37 5.3 Basic Properties for the Generalized Gamma Distribution ...... 38 5.4 Particular Cases and Shapes ...... 39 6 Estimation of parameters ...... 44 6.1 Method of moments ...... 44 6.2 Maximum likelihood estimation ...... 46 7 Product and Ratio of two Generalized Gamma ...... 48 8 Hierarchical Models ...... 51 9 Random Numbers Generation for a Generalized Gamma Distribution ...... 55 9.1 Estimation of the sum of n Generalized Gamma Distributions ...... 57 References ...... 62 10 Appendix ...... 64 10.1 Code to Approximate the sum of k Independent Gamma Random Variables 64 10.2 Code to Approximate the Sum of k Independent Generalized Gamma Ran- dom Variables ...... 75 10.3 Bedfordshire Rainfall Statistics Table ...... 80 Curriculum Vitae ...... 83

vi List of Tables

3.1 Table of Observed and Expected Frequency ...... 18

4.1 Approximation with two moments for the sum of the exact distribution of 2

gamma variables. α1 = 2, α2 = 1.5, β1 = 3, β2 = 2, E(Y ) = 9, V ar(Y ) = 24. 27 4.2 Approximation with two moments for the sum of the exact distribution of 2

gamma variables. α1 = .7, α2 = .5, β1 = 3, β2 = 2, E(Y ) = 3.1, V ar(Y ) = 8.3 28 4.3 Approximation with two moments for the sum of the exact distribution of

5 gamma variables. α1 = 1, α2 = 2, α3 = 3, α4 = 2.5, α5 = 1.2β1 = 2, β2 =

2.5, β3 = 3, β4 = 3.5, β5 = 4, E(Y ) = 29.55, V ar(Y ) = 93.32 ...... 29 4.4 Approximation with two moments for the sum of the exact distribution of

5 gamma variables. α1 = .7, α2 = .3, α3 = .9, α4 = .5, α5 = 1.2β1 = 2, β2 =

2.5, β3 = 3, β4 = 3.5, β5 = 4, E(Y ) = 11.4, V ar(Y ) = 38.1 ...... 30 4.5 Approximation with two moments for the sum of the exact distribution of 10

gamma variables. α1 = 2, α2 = 2.5, α3 = 3, α4 = 3.5, α5 = 4, α6 = 4.5, α7 =

5, α8 = 5.5, α9 = 6, α10 = 6.5β1 = 2, β2 = 2.5, β3 = 3, β4 = 3.5, β5 = 4, β6 =

4.5, β7 = 5β8 = 5.5, β9 = 6, β10 = 6.5, E(Y ) = 201.25, V ar(Y ) = 1030.625. Both graphs are overlapping one to each other due to the good approximation 31 4.6 Normal approximation for the sum of the exact distribution of 2 gamma

variables. α1 = 2, α2 = 1.5, β1 = 3, β2 = 2, E(Y ) = 9, V ar(Y ) = 24. . . . . 32 4.7 Normal approximation for the sum of the exact distribution of 2 gamma

variables. α1 = .7, α2 = .5, β1 = 3, β2 = 2, E(Y ) = 3.1, V ar(Y ) = 8.3 . . . . 33 4.8 Normal approximation for the sum of the exact distribution of 5 gamma

variables. α1 = 1, α2 = 2, α3 = 3, α4 = 2.5, α5 = 1.2β1 = 2, β2 = 2.5, β3 =

3, β4 = 3.5, β5 = 4, E(Y ) = 29.55, V ar(Y ) = 93.32 ...... 34

vii 4.9 Normal approximation for the sum of the exact distribution of 5 gamma

variables. α1 = .7, α2 = .3, α3 = .9, α4 = .5, α5 = 1.2β1 = 2, β2 = 2.5, β3 =

3, β4 = 3.5, β5 = 4, E(Y ) = 11.4, V ar(Y ) = 38.1 Both graphs are overlapping one to each other due to the good approximation ...... 35 4.10 Normal approximation for the sum of the exact distribution of 10 gamma

variables. α1 = 2, α2 = 2.5, α3 = 3, α4 = 3.5, α5 = 4, α6 = 4.5, α7 = 5, α8 =

5.5, α9 = 6, α10 = 6.5β1 = 2, β2 = 2.5, β3 = 3, β4 = 3.5, β5 = 4, β6 =

4.5, β7 = 5β8 = 5.5, β9 = 6, β10 = 6.5, E(Y ) = 201.25, V ar(Y ) = 1030.625 Both graphs are overlapping one to each other due to the good approximation 36

9.1 Empirical distribtuion for a generalized gamma random variable. α = 2, β = 4, δ = 2, E(X) = 5.31, V ar(X) = 3.725 ...... 56 9.2 Normal approximation for the sum of the exact distribution of 2 generalized

gamma variables. α1 = 2, α2 = 5, β1 = 4, β2 = 3, δ1 = 2, δ2 = 3, E(Y ) = 10.33, V ar(Y ) = 4.3 ...... 58 9.3 Normal approximation for the sum of the exact distribution of 2 generalized

gamma variables. α1 = 2, α2 = 5, β1 = 4, β2 = 3, δ1 = .7, δ2 = .8, E(Y ) = 35.40, V ar(Y ) = 326.92 ...... 59 9.4 Normal approximation for the sum of the exact distribution of 5 generalized

gamma variables. α1 = 2, α2 = 5, α3 = 10, α4 = 3, α5 = 78, β1 = 4, β2 =

3, β3 = 81, β4 = 37, β5 = 125, δ1 = 2, δ2 = 3, δ3 = 4, δ4 = 5, δ5 = 6, E(Y ) = 456.012, V ar(Y ) = 188.68 ...... 60 9.5 Normal approximation for the sum of the exact distribution of 10 generalized

gamma variables. α1 = 2, α2 = 5, α3 = 10, α4 = 3, α5 = 7, α6 = .20, α7 =

.9, α8 = 2.5, α9 = .3, α10 = .05β1 = 4, β2 = 3, β3 = 5, β4 = 6, β5 = 2, β6 =

500, β7 = 18β8 = 25, β9 = 20, β10 = 100.5, δ1 = 2, δ2 = 3, δ3 = 4, δ4 = 5, δ5 =

6, δ7 = 8, δ8 = 9, δ9 = 10, δ10 = 11, E(Y ) = 405.1474, V ar(Y ) = 19932.65. . 61

10.1 Rainfall Statistics in mm ...... 80

viii 10.2 Rainfall Statistics in mm (continuation) ...... 81 10.3 Rainfall Statistics in mm (continuation) ...... 82

ix List of Figures

2.1 Example of a standard gamma ...... 8 2.2 Example of an exponential distribution ...... 9 2.3 Example of a chi-square distribution, ν = 5 ...... 10

3.1 Relative Frequency Histogram of Bedfordshire Rainfall ...... 16 3.2 f(x) = 1 x2.617−1e−x/20.9016 ...... 17 Γ(2.619)·(20.99)(2.619) · b 5.1 Weibull, h(x, 2, 2) ...... 40 5.2 Half Normal, g(x, 1) ...... 41 5.3 Analyzing when δ increases ...... 42 5.4 Analyzing when δ decreases ...... 43

9.1 Empirical distribution for a Generalized Gamma ...... 57

x Chapter 1

Introduction

The purpose of this thesis is to review forms following the generalized gamma distribution. Such forms include the exponential, the standard gamma, the weibull and other distributions. These distributions are used in many fields, in particular life studies and they are very common in application of statistics. The generalized gamma distribution is given in a form that has been studied in the literature. We examine the estimation of parameters by the method of maximum likelihood and method of moments. Chapters from two to four concern to the gamma distribution. In chapter two the gamma distribution is briefly studied. How to obtain the density starting from the gamma function and obtaining its mean, variance and moment generating function. Particular cases and shapes are shown, those of the standard gamma, exponential and chi square distribution. For the chi square distribution we give the distribution for a linear combination of random variables each with chi-square distribution (Moschopoulos 1985). In chapter 3 we estimate the parameters for the gamma distribution by using the method of moments and the maximum likelihood estimation. We applied this estimation to show how data from real world can be used to fit a gamma distribution. In chapter four some distributions related to the gamma distribution are presented. We define the log gamma distribution as a random variable X that satis- fies that log(X) follows a gamma distribution. The quotient for two independent gamma distributions is analyzed. The distribution for the sum of k independent gamma when the scale parameters are different is shown (Moschopoulos 1985). We also state two ways to approximate the exact distribution of this sum. One of them is with two moments and the other one is by means of a third moment normal approximation motivated with the ideas of (Jensen and Solomon 1972).

1 Chapters five to nine talk about the generalized gamma distribution. In chapter five the generalized gamma distribution is introduced. We obtain the density for the generalized gamma in the same way as we did for the gamma distribution, by means of a simple change of variable. Some basic properties for the generalized gamma distribution are stated. We obtain the gamma, weibull and half normal distribution as particular cases of the generalized gamma distribution. The shape of the generalized gamma when the parameter δ increases or decreases is obtained and analyzed and this leads to see that δ is a shape parameter. In chapter six we obtain the maximum likelihood estimators for the generalized gamma distribution. Chapter seven talks about the product and ratio of generalized gamma random variables. We start developing the density for the ratio of two random variables having a generalized gamma density. Then we give the distribution for the product of this kind of ratio (Coelho and Joao 2007). Chapter eight gives a new approach for generalized gamma distributions, those of the hierarchical models. We obtain a hierarchical model when Y Λ follows a Poisson distribution and Λ is a generalized gamma random | variable. In chapter nine we obtain a way to generate random numbers for a generalized gamma distribution. We take a look at the empirical distribution of these kind of random variables. And finally we use the same normal approximation as in chapter 4 to estimate the distribution of the sum of k independent generalized gamma random variables.

2 Chapter 2

The Gamma Distribution and its Basic Properties

2.1 History and Motivation

The gamma distribution is known since the Laplace age is mentioned. The gamma distribution appears naturally in theory associated with normally distributed random variables, as the distribution of the sum of squares of independent standard normal variables. In applied work, gamma distributions give useful representations of many physical situations. They have been used to make realistic adjustments to exponential distributions in life- times problems. The model has an application in the theory of random processes in time, in particular in meteorological precipitation processes. Gamma distributions play a very important role when studying the formal theory of mathematical statistics.

2.2 The Gamma Function

The function ∞ Γ(α) = yα−1e−ydy, α > 0, (2.1) Z0 is known as the gamma function. The gamma function satisﬁes the following properties:

1. Γ(α + 1) = αΓ(α)

2. Γ(n) = (n 1)!, where n is a positive integer. − 3. Γ(1/2) = √π.

3 To prove property 1 we have that

∞ ∞ ∞ ∞ Γ(α + 1) = y(α+1)−1eydy = yαeydy = α yαe−y yαe−y , Z Z Z − 0 0 0 0 ∞ using integration by parts. As the term yα−1e−y 0 as y we have that 0 → → ∞

∞ Γ(α + 1) = α yα−1e−y = αΓ(α). Z0 Property 2 follows immediately setting α = n, where n is a positive integer and applying property 1 n times:

Γ(n) = (n 1) Γ(n 1) − × − = (n 1) (n 2) Γ(n 2) − × − × − = (n 1) (n 2) (n 3) Γ(n 3) − × − × − × − . .

= (n 1) (n 2) (n 3) . . . 3 2 1 = (n 1)! − × − × − × × − ∞ −x2 −x2 To prove property 3 we will use the fact that −∞ e dx = π. Now noting that e is R symmetric respect to zero we have ∞ Γ(1/2) = e−xx−1/2dx. Z0 ∞ 2 = e−u du, setting x = u2, we obtain du = x−1/2dx (2.2) Z0 π = . 2 There are interesting alternative ways to deﬁne the gamma function (Continuous Uni- variate Distributions [1]). It can be deﬁned in terms of a limit by

nα Γ(α) = lim . (2.3) n→∞ α α α(1 + α)(1 + 2 ) . . . (1 + n ) Note that this deﬁnition does not involve any integral at all. This deﬁnition is equivalent to 1 n α = αeγα [(1 + )e−α/n], α > 0, n is an integer , (2.4) Γ(α) · n Yi=1

4 where γ is the Euler-Mascheroni constant given by m 1 γ = lim [ log m] 0.5772156649 . . . . (2.5) m→∞ n − ≈ Xn=1 The gamma function is studied in text of advanced calculus and arises often in applications.

2.3 Getting the Gamma Density

A function of the form f(x) = cxα−1e−x/β, α, β > 0, (2.6) deﬁnes a very special probability density. To ﬁnd c we set ∞ cxα−1e−x/βdx = 1 Z0 Letting y = x/β the integral above becomes ∞ ∞ cxα−1e−x/βdx = cβα yα−1e−ydy = cβαΓ(α) = 1, Z0 Z0 from where we get 1 c = . βαΓ(α) Now with this value for c the gamma density in (2.6) becomes 1 f(x, α, β) = xα−1e−x/β, α, β > 0. (2.7) βαΓ(α)

2.4 Mean, Variance and Moment Generating Func- tion

We will use the notation X G(α, β) to denote that a random variable X with density ∼ given by (2.7), follows a gamma distribution with parameters α, β. The rth moment for the gamma distribution is given by βrΓ(α + r) µ0 = E(Xr) = , α, β > 0, r a positive integer. (2.8) r Γ(α)

5 The derivation for the rth moment comes from the deﬁnition ∞ r 1 α−1 r −x/β E(X ) = α x x e dx β Γ(α) Z0 · ∞ 1 α+r−1 −x/β = α x e dx β Γ(α) Z0 1 βrΓ(α + r) = Γ(α + r)βα+r = . βαΓ(α) Γ(α) Thus we have that the mean is equal to βΓ(α + 1) βαγ(α) µ0 = E(X) = = = αβ. (2.9) 1 Γ(α) γ(α) The variance is given by

2 β Γ(α + 2) βΓ(α + 1) 2 V ar(X) = µ0 (µ0 )2 = = αβ2. (2.10) 2 − 1 Γ(α) − Γ(α) The moment generating function for the gamma is given by ∞ tX tx 1 α−1 −x/β MX (t) = E(e ) = e α x e dx, < t < . (2.11) Z0 · β Γ(α) − ∞ ∞ In order to get this moment generating function we will try to express the right side above as a gamma density. We do the change of variable y = x(1 βt) then: − ∞ tX 1 α−1 −x(1−βt)/β E(e ) = α x e dx Z0 β Γ(α) ∞ α−1 1 y 1 −y = α α−1 e dy Z0 β Γ(α) (1 βt) · 1 βt − − ∞ 1 1 1 1 = yα−1e−ydy = . (1 βt)α−1 · 1 βt · βαΓ(α) Z (1 βt)α − − 0 − One application for the moment generating function of a gamma distribution is that of the reprodcutive property. If X G(α , β) and X G(α , β) are independent with 1 ∼ 1 2 ∼ 2 respective densities

1 α1−1 −x/β f1(x, α1, β) = α x e , α1, β > 0 (2.12) β 1 Γ(α1) and

1 α2−1 −x/β f2(x, α2, β) = α x e , α1, β > 0, (2.13) β 2 Γ(α2)

6 then X + X G(α + α , β). That is, the density for the sum of the random variables 1 2 ∼ 1 2 X1 + X2 is

1 α1+α2−1 −x/β fX1+X2 (x, α1 + α2, β) = α +α x e . (2.14) β 1 2 Γ(α1 + α2) This easily seen using the properties for the moment generating function. Note that M (t) = M (t) M (t) because X and X are independent. So we have X1+X2 X1 · X2 1 2 − − − M (t) = M (t) M (t) = (1 βt) α1 (1 βt) α2 = (1 βt) (α1+α2). X1+X2 X1 · X2 − · − − Here we use the fact that there exists a unique correspondence between moment generating function and distribution. This means that as MX1+X2 (t) has the form of a moment generating function for a gamma with parameters β, α1 + α2 then X1 + X2 has a gamma distribution with parameters β and α1 + α2. We can even generalize this to k independent gamma’s. Let Xi, . . . Xk be k independent random variables each one following a gamma distribution with parameters (αi, β), i = 1, 2, . . . k. Then the moment generating function for these k independent gamma is

−α −α − Pk α M · · (t) = M (t) . . . M (t) = (1 βt) 1 . . . (1 βt) k = (1 βt) i=1 i . X1 ... Xk X1 · Xk − · · − −

k This is the moment generating function for a gamma with parameters i=1 αi and β. So by the same reasoning as in the two parameters case we conclude that thePrandom variable k X1 + . . . Xn has a gamma distribution with parameters ( i=1 αi, β). P

2.5 Particular Cases and Shapes

2.5.1 Standard Gamma

Several forms of the gamma density can be obtained by changing the parameters in (2.7). We will analyze these forms and study them for some values. One of them is obtained by setting β = 1 in (2.7). That is 1 f(x, α, 1) = xα−1 e−x, x > 0, α > 0, (2.15) Γ(α) ·

7 this form for the gamma is called the standard gamma. The integral

x α−1 t Γx(α) = t e dt Z0 is called the incomplete gamma function.

Gamma distribution

alpha_1=2,beta_1=1 0.3 0.2 y

alpha_2=2,beta_2=1 0.1 0.0

0 2 4 6 8 10

Figure 2.1: Example of a standard gamma

2.5.2 The Exponential

Another very important form of the gamma distribution is obtained by setting α = 1 in (2.7). This is the exponential distribution.

f(x, 1, β) = e−x/β, x > 0, β > 0. (2.16)

This distribution arises often in practice. It can be used to model the waiting time between arrivals of a Poisson process (see [8] pp 203). According to (2.9) and (2.10) we can see that the the mean and variance for the exponential are equal to β and β2 respectively.

8 Gamma distribution

red.−alpha_1=1,beta_1=1 0.20 0.15 y 0.10 blue.−alpha_2=1,beta_2=2 0.05 0.00 2 4 6 8 10 12

Figure 2.2: Example of an exponential distribution

2.5.3 The Chi square distribution

A chi-square distribution is perhaps the most important special case of a gamma distribution. This distribution is deﬁned with α = ν/2, ν > 0 and β = 2.

1 f(x, ν/2, 2) = x(ν/2)−1e−x/2 , x > 0, ν > 0. (2.17) 2ν/2Γ(ν/2)

We then say that the density in (2.17) is a Chi-square with ν degrees of freedom. If a random variable X follows a chi-square distribution with ν degrees of freedom we will denote it as X χ2(ν). This distribution plays a very important role in many applications. It also ∼ serves as a way to characterize the square of standard normal distributions. For example if U1, U2 . . . Un are standard normal random variables (N(0, 1)) then the distribution of n 2 i=1 Ui has a chi-square distribution with n degrees of freedom. P The mean and variance of the chi square χ2(ν) distribution are ν and 2ν respectively.

9 Gamma distribution 0.20 alpha_1=5/2,beta_1=2 y 0.10 0.00 0 5 10 15 20

Figure 2.3: Example of a chi-square distribution, ν = 5

2.5.4 Linear Combinations of Independent Chi-Square

We include here the distribution of a linear combination of independent chi-square ([10] Moschopoulos 1985), X χ2(n ) deﬁned by i ∼ i p 2 Y = ciχ (ni), ci > 0, ni a positive integer for i = 1, 2, 3 . . . p (2.18) Xi=1 2 where χ (ni), i = 1 . . . p is independent with ni degrees of freedom. The moment generating function for the equation in (2.18) is obtained as p − M(t) = (1 2c t) mi , m = n /2. (2.19) − i i i Yi=1

For t < min(1/2ci) then ∞ − (m ) − (1 2c t) mi = (c /c )mi i r (1 c /c )r(1 2c t) (r+mr). (2.20) − i 1 i r! − 1 i − i Xr=0 So that with this information and multipliying the expression in (2.19) we get p ∞ p M(t) = b a (1 2c t)−(s+j) , s = m , (2.21) i j − 1 i Xi=2 Xj=0 Xi=1

10 with b = (c /c )mi and A(i, r) = (m ) (1 c /c )r/r!, i 1 i i r − 1 i besides the ajs satisfy the relation

p ∞ ∞ −r −j [ A(ci, r)x ] = ajx . Yi=2 Xr=0 Xj=0

From this we can get the ajs recursively that is

j a = A(p), A(i) = Ai−1A(c , j k) (2.22) j j j k i − Xk=0 where i = 3, 4, . . . , p j = 0, 1, 2 . . . and for r = 0, 1, 2, . . .

(2) Ar = A(C2, r).

And now we invert (2.21) term by term. Since the density corresponding to the factor (1 2c t)−(s+j) is the gamma density g (y) where − 1 j

s+j−1 −y/2c1 s+j gj(y) = y e /(2c1) Γ(s + j) the density function F (w) = P (Y w) (the probability that the random variable Y is less ≤ or equal than w) is p ∞ ∞

F (w) = bi aj gj(y)dy. (2.23) Z Yi=2 Xj=0 0 ([10] Moschopoulos 1985).

11 Chapter 3

Estimation of Parameters

3.1 Method of Moments

We know from (2.8) that the r-th moment for X G(α, β) is given by ∼ βrΓ(α + r) µ0 = , r Γ(α) these are the population moments for r = 1, 2, 3 . . ..

If we consider random variables X1, X2, . . . Xn G(α, β), and if we observe a particular ∼ n xr value from each, x , x , . . . x , then the sample moments are given by i=1 i . The method 1 2 n P n of moments consists on setting those population moments equal to the sample moments. Considering the cases r = 1, 2, we must solve the equations

βΓ(α + 1) n x = i=1 i (3.1) Γ(α) P n and β2Γ(α + 2) n x2 = i=1 i (3.2) Γ(α) P n for β and α. By using the properties for the gamma function (3.1) and (3.2) become

n x αβ = i=1 i , (3.3) P n n x2 β2(α + 1)α = i=1 i . (3.4) P n x n x Solving (3.3) for β we obtain β = , where x = i=1 i . Now if we substitute in (3.4) for α P n

12 α we have n x2 β2(α + 1)α = i=1 i P n x2 n x2 x (α + 1)α = i=1 i ( substituing β = ) ⇔ α2 P n α x2(α + 1) n x2 = i=1 i ⇔ α P n (α + 1) n x2 = i=1 i ⇔ α Px2n 1 n x2 1 + = i=1 i ⇔ α Px2n 1 x2 x2 x2n = i 1 = i − , ⇔ α Pxn − P x2n from where ﬁnally we get x2n α = x2 x2n i − and b P x β = α b b 3.2 Maximum Likelihood Estimation

Let X , X , . . . X G(α, β) for i = 1, 2 . . . n. The likelihood function is given by 1 2 n ∼ n 1 α−1 − Pn (x /β) L(x , . . . x , α, β) = f(x , α, β) = (x , . . . x ) e i=1 i . (3.5) 1 n i Γ(α)n βnα 1 n Yi=1 · The maximum likelihood estimation method consists on ﬁnding the values for α and β such that (3.5) is a minimum. Taking natural log at both sides of (3.5) and using the properties for logarithms we have

ln(L(x . . . x , α, β)) = n ln(Γ(α)) nα ln(β) + (α 1) ln(x , . . . x ) Σn (x /β). (3.6) 1 n − − − 1 n − i=1 i The derivative respect to α in (3.6) set it equal to zero give the corresponding equation

∂L nΓ0(α) n = n ln(β) + ln(x ) = 0. ∂α − Γ(α) − i Xi=1

13 This can be expressed as

∂L nΓ0(α) n = + ln(x ) ln(β) = 0 ∂α − Γ(α) i − Xi=1 (3.7) nΓ0(α) n = + ln(x /β) = 0. − Γ(α) i Xi=1 Diﬀerentiating with respect to β

∂L nα n x = − + i=1 i . (3.8) ∂β − β Pβ2

Setting (3.8) equal to zero the equation is

nα n x − + i=1 i = 0 − β Pβ2 n x nα + i=1 i = 0 ⇔ − P β n x i=1 i = nα. ⇔ P β From where we obtain n x x β = i=1 i = . (3.9) Pnα α Where we have used the notation β bto denote the maximum likelihood estimator for β. Try to solve for α the equation b(3.7) is quite complicated because of the function Γ(α). There is no closed way to solve for α in this equation. By adding the value for β in terms of α obtained in (3.9) to (3.7) we get

nΓ0(α) n nΓ0(α) n + ln(x /β) = + ln(x α/x) − Γ(α) i − Γ(α) i Xi=1 Xi=1 nΓ0(α) n = n + ln(x ) + ln(α) ln(x) − Γ(α) i − Xi=1 n (3.10) = nψ(α) + ln(x ) + n ln(α) n ln(x) − i − Xi=1 n ln(x ) = ψ(α) + i=1 i + ln(α) ln(x) = 0 − P n − n ln(x ) ln(α) ψ(α) = ln(x) i=1 i , ⇔ − − P n

14 Γ0(α) where ψ(α) is the digamma function deﬁned as ψ(α) = Γ(α) . One approximation for the left side of (3.10) is 1 1 1 ln(α) ψ(α) + . − ≈ α 2 12α + 2 So (3.10) becomes 1 1 1 n ln(x ) + ln(x) i=1 i . α 2 12α + 2 ≈ − P n Finally solving this equation for α we obtain the approximate maximum likelihood estimation for α 3 s + (s 3)2 + 24s α − − (3.11) ≈ p12s where s = ln(x) 1 n ln(x b) ([5] S. C. Choi and R. Wette. (1969)). − n i=1 i P

3.3 Applications

Now we will use the previous estimators to ﬁt a gamma density to a set of data. Data set 1 The Bedfordshire county give the next information about the daily rainfall since january 1984 up to december 2010 by month (see Table 10.1). The histogram for this data is given in ﬁgure ( 3.1). The estimation of the parameters by means of the maximum likelihood is given by (3.11) 3 s + (s 3)2 + 24s α − − ≈ p12s where s is given by ln(x) 1 b n ln(x ) and − n i=1 i P x β = . α b The estimation of the parameters lead to α 2.617 and β = 20.9016. So that the estimated ≈ density function is given by b b 1 f(x) = Γ(2.617) x2.617e−x/20.9016 (20.9016)2.617 · · b

15 Bedfordshire Rainfall 0.012 0.008 Density 0.004 0.000 0 50 100 150

data

Figure 3.1: Relative Frequency Histogram of Bedfordshire Rainfall

At first glance it seems to be a very good fit to this data in figure ( 3.2), where we have the estimated density with the estimated parameters as above. We can be sure of this by doing a goodness of fit test. With the hypotheses

H0 : Data follows a gamma distribution.

Ha : Data doesn’t follow a gamma distribution.

The used statistic is 9 (f e )2 χ2 = i − i , e Xi=1 i The data in table (10.1) was arranged into nine class intervals. The table shows 324 numbers representing rainfall in milimeters. We omit the value for august in 2003 because it is zero. Here fi and ei are the observed and expected frequency for each class respectively, for i = 1, 2, . . . , 9. Under the hypothesis that H0 is true the expected frequency is deﬁned as e = 323 f(y ) f(y ) , i = 0, 2, . . . 8 and y = 0, y = 20, y = 40, . . . y = 180. i+1 × i+1 − i 0 1 2 9 This information isb summarizedb in table (3.3). 2 2 We reject H if χ χ − − , otherwise we do not reject H . Here m is the number of 0 ≥ m t 1 0

16 Bedfordshire Rainfall

Relative frequency histogram 0.012

Estimated density 0.008 Density 0.004 0.000 0 50 100 150

data

Figure 3.2: f(x) = 1 x2.617−1e−x/20.9016 Γ(2.619)·(20.99)(2.619) · b terms in the summation and t is the number of parameters estimated. In this case m = 9 and t = 2. Hence we have that

2 2 χ = 3.0421 < χ6 = 12.5916,

so that we do not reject H0 and conclude that this data does follow a gamma distribution.

17 Table 3.1: Table of Observed and Expected Frequency

Class Interval Observed Frequency Expected Frequency 0-20 38 38.76 20-40 90 88.43 40-60 80 79.40 60-80 51 53.18 80-100 29 30.88 100-120 18 16.49 120-140 9 8.32 140-160 7 4.04 160-180 1 1.903

18 Chapter 4

Distributions Related to the Gamma Distributions

4.1 Log Gamma Distribution

We say that a random variable X is log gamma distributed if ln(X) is gamma distributed. This means that if Y = ln(X) then the density for Y denoted by f(y, α, β) is given by 1 f (y, α, β) = yα−1e−y/β, β, α, y > 0. (4.1) Y βα Γ(α) · We will get the density for X. Noting that Y is a transformation for X the density for X is dy dy 1 given by f(x, α, β) = f (ln(x)) . As we know that ln(x) = y and = we obtain Y · dx dx x

(ln (x ))α−1 f(x, α, β) = e− ln(x)/β xβαΓ(α) · (ln(x))α−1 (ln(x))α−1 = x−1/β = x−(β+1)/β, x > 1. xβαΓ(α) · βαΓ(α) · Some properties ([6] P.C Consul and G. C. Jain) for the log gamma distribution are:

1. If Y is a log gamma with density f(x, α, β) then Y n is also a log gamma distribution with density f(x, α/n, β).

2. If Y1, Y2, . . . Yn are independent log gamma variables with densities f(xi, α, βi) for n n i = 1, 2 . . . , then U = Πi=1Yi is also a log gamma density with density f(u, α, i=1 βi). P 3. If Y1, Y2, . . . Yn is a sequence of independent log gamma variables, where Yi has density n n f(xi, α, βi) for all i = 1, 2 . . . , then the product U = Πi=1Yi is also a log gamma n variable with density f(u, α/n, i=1 βi). P 19 4.2 The Quotient of Two Independent Gamma

Now if we consider the ratio X/Y , where X G(α , β ) and Y G(α , β ) and they are ∼ 1 1 ∼ 2 2 independent. Setting X u = X + Y, v = , Y − − 2 then the jacobian becomes J = (1 v) . So | | u

1 uv α −1 u α −1 u 2 f (u, v) = e−u 1 2 U,V Γ(α )Γ(α ) 1 + v 1 + v 1 + v 1 2 1 − − − − = e uuα1+α2 1vα2 1(1 + v) (α1+α2). Γ(α1)Γ(α2) If we integrate this with respect to u we obtain ∞ 1 −u α1+α2−1 α2−1 −(α1+α2) fV (v) = e u v (1 + v) du Γ(α1)Γ(α2) Z0 ∞ 1 − − − − = vα2 1(1 + v) (α1+α2) e uuα1+α2 1du Γ(α1)Γ(α2) Z0

1 α2−1 −(α1+α2) = v (1 + v) Γ(α1 + α2), Γ(α1)Γ(α2) ∞ −u α1+α2−1 because e u du = Γ(α1 + α2). Z0 X 4.3 The Quotient 1 X1 + X2

If X1 and X2 are independent random gamma variables with parameters (α1, β) and (α2, β) respectiveley, then X1/(X1 + X2) has a beta distribution with parameters α1, α2. We can deduce this from 1 − − f(x ) = xα1 1e x/β, 1 βα1 Γ(α ) 1 1 · 1 1 − − f(x ) = xα2 1e x/β, 2 βα2 Γ(α ) 1 2 · 2 then as the random variables are independent we have

1 1 1 −(x1+x2) − − β α1 1 α2 1 f(x1, x2) = e x1 x2 . (4.2) Γ(α ) Γ(α ) βα1 · βα2 1 · 2

20 Now we set

u = X1 + X2, v = X1/(X1 + X2), then X = uv, X = u(1 v). 1 2 − With this the jacobian becomes J = u. So | | u − − − − f (u, v) = e u/β (uv)α1 1uα2 1(1 v)α2 1 U.V Γ(α )Γ(α ) · − 1 2 (4.3) u − − − − = e u/βuα1+α2 1vα1 1(1 v)α2 1. Γ(α1)Γ(α2) − Therefore, integrating (4.3) respect to v the sum has distribution

− − e u/βuα1+α2 1 f(u) = , Γ(α1 + α2) which is a gamma distribution with parameters α1 + α2 and β. Now using (4.3) and integrating respect to u the ratio v = X1/(X1 + X2) has distribution

Γ(α + α ) − − f (v) = 1 2 vα1 1 (1 v)α2 1, V Γ(α ) Γ(α ) · − 1 · 2 which is a beta function with parameters α1, α2.

4.4 Sum of k Independent Gamma

k We can say even more about the random variable Y = i=1 Xi where each variable has gamma distribution, that is X (α , β ), i = 1, 2 . . . nPand they are independent ([9], i ∼ i i Moschopoulos 1985) . We can obtain the exact distribution for Y. We know that each Xi has moment generating function given by

− M(t) = (1 β t) αi , − i then Y has moment generating function given by

k − M(t) = (1 β t) αi . − i Yi=1

21 The idea is to express M(t) as

∞ M(t) = C(1 β t)−ρ exp( γ (1 β t)−k), − i · k − i Xk=1 where exp(a) = ea and

αi C = (β1/βi) , Y 0 with β1 = min(β1, β2 . . . βn) (the minimum among all β s),

n γ = α (1 β /β )k/k, k i − 1 i Xi=1

ρ = αi > 0. Xi=1

And now if we let ∞ ∞ exp( γ (1 β t)−k) = δ (1 β t)−k. k − i k − 1 Xk=1 Xk=0 By diﬀerentiating with respect to (1 βt)−1, the coeﬃcients δ can be obtained by means − k of the formula 1 k+1 δ = iγ δ − . k+1 k + 1 i k+1 i Xi=1 This leads to the next theorem which gives an expression for Y ([9] Moschopoulos 1985).

Theorem 1. If X (α , β ) and independently distributed for i = 1, 2, . . . n. Then the i ∼ 1 i density for Y can be expressed as

∞

ρ+k−1 −y/β1 ρ+k g(y) = C δky e /[Γ(ρ + k)β1 ], y > 0 (4.4) Xk=0 and 0 elsewhere.

22 4.4.1 Aprroximation for the Exact Distribution of the sum of k Independent Gamma

Approximation with Two Moments

If X G(α , β ) for i = 1 . . . n, and Y = X + . . . + X . Then we know that the density is i ∼ i i 1 n given by (4.4) ∞

ρ+k−1 −y/β1 ρ+k g(y) = C δky e /[Γ(ρ + k)β1 ], y > 0. (4.5) Xk=0 If we want to compute G(y), the distribution for this sum, we need to evaluate this iniﬁnite sum. But we can approximate it assuming that Y G(α, β) and knowing two ∼ moments of this assumed distribution. If Y G(α, β) then EY = αβ and V ar(Y ) = αβ 2, ∼ from where we obtain V ar(Y ) β = , E(Y ) and E(Y )2 α = , V ar(Y ) 2 2 but E(Y ) = α1β1 + α2β2 + . . . αnβn and V ar(Y ) = α1β1 + . . . + αnβn. So that with these values we obtain assuming that Y G(α, β) ∼ α β2 + . . . + α β2 β = 1 1 n n α1β1 + α2β2 + . . . αnβn and 2 (α1β1 + α2β2 + . . . αnβn) α = 2 2 . (α1β1 + . . . + αnβn) If we use this values for α and β we can approximate (4.5) with G(α, β), where α and β are given as above.

Normal Approximation

If Xi, i = 1 . . . n are independent standard normal variables and considering the following linear combination ([18] Solomon and Jensen 1972) k Q (c, a) = c (x2 + a ), c > 0, 1 j k, (4.6) k j j j j ≥ ≤ Xj=1

23 then the sth cumulant of Qk(c, a) is

k κ = 2s−1(s 1)! cs(1 + sa2). s − j j Xj=1

Under the condition that the parameters cj, aj are bounded, it can be shown that the distribution of Q tends to a gaussian distribution as θ where θ = E(Q (c, a)). So k 1 → ∞ 1 k h following this idea (Solomon and Jensen 1972) used the transformation (Qk/θ1) to accel- erate the rate of convergence and thus to provide a Gaussian approximation for moderate

h −1 values of θ1. The moments of (Qk/θ1) expanded in powers of (θ1) were obtained. In particular the third central moment is 4h2 µ (h) = 2φ + 3(h 1)φ2 3 θ2 3 − 2 1 h2(h 1) − 72φ + 24(7h 10)φ φ (4.7) θ3 4 − 2 3 1 + 4(17h2 55h + 44)φ2 + 0(θ−4), − 2 1 h where φr = θr/θ1, r = 2, 3. A similar expression for γ1, the skewness of (Qk/θ1) is

2 4 2φ3 + 3(h 1)φ γ (h) = − 2 1 1/2 3/2 θ1 (2φ2 ) 4(17h2 55h + 44)φ3 (h 1) 72φ + 24(7h 10)φ φ + − 2 (4.8) · − 4 − 2 3 5/2 θ1 −5/2 + 0(θ1 ).

Now h is chosen so that the leading term in γ1 and µ3 vanishes. This is

2θ θ 2 3 + 3(h 1) 2 = 0. (4.9) θ − θ 1 1 This yields to 2θ1θ3 h = 1 2 . (4.10) − 3θ2 h Now the distribution of the random variable (Qk/θ1) can be approximated by a Gaussian distribution with mean µ0 (h) = 1 + θ h(h 1)/θ2, (4.11) 1 2 − 1

24 and variance 2 h V ar(Qk/θ1) (h) = 2θ2h /θ1. (4.12)

In summary the variable z = θ (Q /θ )h 1 θ h(h 1)/θ2 /(2θ h2) is approximately a 1 k 1 − − 2 − 1 2 standard normal variable. Now, motivated with this ideas, we will consider another approximation for the distribution of Y = X +X +. . . X , with X G(α , β ), i = 1, 2 . . . n. We propose that (Y/θ )h 1 2 n i ∼ i i 1 with h given as in (4.10) and E(Y ) = θ1, can be approximated with a standard normal 2 distribution with mean µh and variance σh given as follows. We introduce the cumulant κn of the random variable Y deﬁned as

dn κ = g(n)(t) = g(t), n dt where ∞ 1 n g(t) = ln E(etY ) = 1 E(etY ) − n − Xn=1 ∞ ∞ m 1 t n = µ0 n − m m! Xn=1 mX=1 t2 t3 = µ0 t + µ0 µ0 + µ0 3µ0 µ0 + 2(µ0 )3 + . . . . 1 2 − 1 2! 3 − 2 1 1 3! ∞ − n We have used the fact that ln(x) = (1 x) − n=1 n Hence we have that P 0 0 κ1 = g (0) = µ ,

00 0 0 2 κ2 = g (0) = µ2 + (µ1) ,

κ = g(3)(0) = µ0 3µ0 µ0 + 2(µ0 )3. 3 3 − 2 1 1 (n) In general κn = g (0). Now if we deﬁne

h = 1 κ κ /(3 κ2), − 1 3 · 1 h(h 1) φ h (h 1) (h 2) (4 φ + 3 (h 3) φ2) µ = 1 + − · 2 + · − · − · · 3 · − · 2 h 2 κ 24 κ2 · 1 · 1

25 and 2 2 2 2 φ2h (h 1) h 2φ3 + (3h 5)φ2 σh = + − · 2 − , κ1 2κ1 with κ2 φ2 = , κ1 κ3 φ3 = , κ1 then we take (Y/θ )h µ Z = 1 − h (4.13) σ h to be approximately a standardized Gaussian variable. So that we can use this information to approximate the distribution of Y. This kind of approximation has been used in literature with other applications. Moschopoulos in 1983 ([11]) and Moschopoulos and Mudholkar in 1983 ([12]) used and actually improved the eﬃciency of these approximations. In the next pages tables with the normal and two moments approximation are shown. We consider here the distribution P (Y t) where Y = X + . . . + X . ≤ 1 n

26 Table 4.1: Approximation with two moments for the sum of the exact distribution of 2 gamma variables. α1 = 2, α2 = 1.5, β1 = 3, β2 = 2, E(Y ) = 9, V ar(Y ) = 24.

t Exact Approximate Diﬀerence 5 .2122 .2137 -.0014 6 .3045 .3050 -.0005 7 .3989 .3985 .0003 8 .4899 .4888 .0011 9 .5740 .5724 .0015 10 .6490 .6472 .0017 11 .7143 .7125 .0017 12 .7698 .7682 .0015

Exact and Two Moments App Distribution 1.0 0.8 Red=Exact 0.6 Ex 0.4 Blue=Two Mome App 0.2 0.0

0 5 10 15 20 25 30

27 Table 4.2: Approximation with two moments for the sum of the exact distribution of 2 gamma variables. α1 = .7, α2 = .5, β1 = 3, β2 = 2, E(Y ) = 3.1, V ar(Y ) = 8.3

t Exact Approximate Diﬀerence 1 .2404 .2442 -.0038 3 .6110 .6099 .0011 5 .8070 .8051 .0019 7 .9051 .9040 .0010 9 .9533 .9531 .0002 11 .9770 .9772 -.0001 13 .9887 .9889 -.0002 15 .9944 .9946 -.0002

Exact and Two Moments App Distribution 1.0

Red=Exact 0.8 Ex 0.6

Blue=Two Mome App 0.4

0 5 10 15 20 25 30

28 Table 4.3: Approximation with two moments for the sum of the exact distribution of 5 gamma variables. α1 = 1, α2 = 2, α3 = 3, α4 = 2.5, α5 = 1.2β1 = 2, β2 = 2.5, β3 = 3, β4 = 3.5, β5 = 4, E(Y ) = 29.55, V ar(Y ) = 93.32

t Exact Approximate Diﬀerence 20 .1558 .1565 -.0007 23 .2661 .2661 -.0000 26 .3931 .3924 .0006 29 .5219 .5207 .0011 32 .6401 .6388 .0012 35 .7400 .7390 .0010 38 .8191 .8185 .0006 41 .8783 .8781 .0002 44 .9206 .9206 .0000

Exact and Two Moments App Distribution 0.8 Red=Exact 0.6 Ex

0.4 Blue=Two Mome App 0.2 0.0

0 10 20 30 40

29 Table 4.4: Approximation with two moments for the sum of the exact distribution of 5 gamma variables. α1 = .7, α2 = .3, α3 = .9, α4 = .5, α5 = 1.2β1 = 2, β2 = 2.5, β3 = 3, β4 = 3.5, β5 = 4, E(Y ) = 11.4, V ar(Y ) = 38.1

t Exact Approximate Diﬀerence 12 .6112 .6088 .0023 14 .7184 .7161 .0023 16 .8013 .7994 .0018 18 .8627 .8615 .0011 20 .9067 .9062 .0005 22 .9375 .9374 - .0000 24 .9586 .9589 - .0002 26 .9728 .9733 - .0004 28 .9886 .9828 - .0004

Exact and Two Moments App Distribution 1.0 0.8 Red=Exact 0.6 Ex 0.4

0.2 Blue=Two Mome App 0.0

0 10 20 30 40

30 Table 4.5: Approximation with two moments for the sum of the exact distribution of 10 gamma variables. α1 = 2, α2 = 2.5, α3 = 3, α4 = 3.5, α5 = 4, α6 = 4.5, α7 = 5, α8 = 5.5, α9 = 6, α10 = 6.5β1 = 2, β2 = 2.5, β3 = 3, β4 = 3.5, β5 = 4, β6 = 4.5, β7 = 5β8 = 5.5, β9 = 6, β10 = 6.5, E(Y ) = 201.25, V ar(Y ) = 1030.625. Both graphs are overlapping one to each other due to the good approximation

t Exact Approximate Diﬀerence 180 .2636 .2633 .003 200 .5068 .5056 .0011 220 .7330 .7322 .0007 240 .8828 .8829 -.0000 260 .9576 .9581 -.0004 280 .9868 .9875 - .0006 300 .9958 .9968 - .0009 320 .9980 .9993 - .0012 340 .9984 .9998 - .0013

Exact and Two Moments App Distribution 1.0 Red=Exact 0.8 0.6

Ex Blue=Two Mome App 0.4 0.2 0.0

0 50 100 150 200 250 300

31 Table 4.6: Normal approximation for the sum of the exact distribution of 2 gamma variables. α1 = 2, α2 = 1.5, β1 = 3, β2 = 2, E(Y ) = 9, V ar(Y ) = 24.

t Exact Approximate Diﬀerence 5 .2122 .2108 .0014 6 .3045 .3030 .0015 7 .3989 .3977 .0011 8 .4899 .4892 .0006 9 .5740 .5738 .0001 10 .6490 .6493 -.0003 11 .7143 .7149 -.0006 12 .7698 .7706 -.0008

Exact and Normal App Distribution 1.0 0.8 Red=Exact 0.6 Ex 0.4 Green=Normal App 0.2 0.0

0 5 10 15 20 25 30

32 Table 4.7: Normal approximation for the sum of the exact distribution of 2 gamma variables. α1 = .7, α2 = .5, β1 = 3, β2 = 2, E(Y ) = 3.1, V ar(Y ) = 8.3

t Exact Approximate Diﬀerence 1 .2404 .2358 .0045 3 .6110 .6100 .0009 5 .8070 .8091 -.0021 7 .9051 .9069 -.0018 9 .9533 .9543 -.0009 11 .9770 .9773 -.0003 13 .9887 .9886 .0000 15 .9944 .9942 .0001

Exact and Normal App Distribution 1.0

Red=Exact 0.8 Ex 0.6

Green=Normal App 0.4

0 5 10 15 20 25 30

33 Table 4.8: Normal approximation for the sum of the exact distribution of 5 gamma variables. α1 = 1, α2 = 2, α3 = 3, α4 = 2.5, α5 = 1.2β1 = 2, β2 = 2.5, β3 = 3, β4 = 3.5, β5 = 4, E(Y ) = 29.55, V ar(Y ) = 93.32

t Exact Approximate Diﬀerence 20 .1558 .1554 -.0003 23 .2661 .2656 -.0004 26 .3931 .3924 .0003 29 .5219 .5218 .0000 32 .6401 .6403 -.0001 35 .7400 .7404 -.0003 38 .8191 .8195 -.0003 41 .8783 .8786 -.0002 44 .9206 .9207 -.0001

Exact and Normal App Distribution 0.8 Red=Exact 0.6

Ex Green=Normal App 0.4 0.2 0.0

0 10 20 30 40

34 Table 4.9: Normal approximation for the sum of the exact distribution of 5 gamma variables. α1 = .7, α2 = .3, α3 = .9, α4 = .5, α5 = 1.2β1 = 2, β2 = 2.5, β3 = 3, β4 = 3.5, β5 = 4, E(Y ) = 11.4, V ar(Y ) = 38.1 Both graphs are overlapping one to each other due to the good approximation

t Exact Approximate Diﬀerence 12 .6112 .6114 -.0001 14 .7184 .7191 -.0006 16 .8013 .8021 -.0008 18 .8627 .8634 -.0007 20 .9067 .9073 -.0005 22 .9375 .9378 -.0003 24 .9586 .9588 - .0001 26 .9728 .9729 - .0000 28 .9886 .9823 .0000

Exact and Normal App Distribution 1.0 0.8

0.6 Green=Normal App Ex 0.4

Red=Exact 0.2 0.0

0 10 20 30 40

35 Table 4.10: Normal approximation for the sum of the exact distribution of 10 gamma variables. α1 = 2, α2 = 2.5, α3 = 3, α4 = 3.5, α5 = 4, α6 = 4.5, α7 = 5, α8 = 5.5, α9 = 6, α10 = 6.5β1 = 2, β2 = 2.5, β3 = 3, β4 = 3.5, β5 = 4, β6 = 4.5, β7 = 5β8 = 5.5, β9 = 6, β10 = 6.5, E(Y ) = 201.25, V ar(Y ) = 1030.625 Both graphs are overlapping one to each other due to the good approximation

t Exact Approximate Diﬀerence 180 .2636 .2635 .0000 200 .5068 .5068 .0000 220 .7330 .7331 -.0000 240 .8828 .8829 -.0000 260 .9576 .9577 -.0001 280 .9868 .9871 - .0003 300 .9958 .9966 - .0007 320 .9980 .9992 - .0011 340 .9984 .9998 - .0013

Exact and Normal App Distribution 1.0 Red=Exact 0.8 0.6 Green=Normal App Ex 0.4 0.2 0.0

0 50 100 150 200 250 300

36 Chapter 5

Generalized Gamma Distributions

5.1 History

Generalized gamma distributions were discussed since the beginning of the 20th century. It appeared when ﬁtting such distribution to an observed distribution of income rates. There was not a good interest in this kind of distribution up to the middle of the 20th century. From that time on, the interest in the generalized gamma was increasing. Properties, applications, distributions which were related to a generalized gamma were showing up. Stacy (see [15] 1963) published a paper which developed the idea of a generalized gamma introducing basic properties. It was the ﬁrst paper talking entirely about this generalized distribution.

5.2 Motivation and Getting the Generalized Gamma Density

So far we have seen several properties for the gamma distribution. We will focus our attention on investigating the properties of the generalized gamma functions. If X is a random variable we say X has generalized gamma density if its pdf is given by

− −( x )δ f(x) = cxδα 1e β , x, α, β, δ > 0. (5.1)

This distribution contains all previous densities we have just seen, such as gamma density and its densities derived from it. This is one form of the generalized gamma

37 that appears often in literature (see [1]), but there exist others forms that depend on the parameters. This is form is convenient for our purposes. The constant c in (5.1) is the reciprocal of the integral ∞ − −( x )δ xδα 1e β dx. Z0

δ δα−1 α− 1 δα−1 If we let y = (x/β) we get x = y δ β . And the diﬀerential dx is given by · 1−δ dx = (β/δ) y δ . · And then we get ∞ ∞ ∞ δα−1 −( x )δ δα 1 α− 1 + 1 −1 −y δα 1 α−1 −y cx e β dx = β cy δ δ e dy = β cy e dy = 1. Z0 · δ Z0 · δ Z0 But as ∞ yα−1e−ydy Z0 has the form of a standard gamma we obtain ∞ yα−1e−ydy = Γ(α). Z0 Finally have 1 ∞ 1 βδα cyα−1e−ydy = βδα cΓ(α) = 1. · δ Z0 · δ From where we get δ c = . βδαΓ(α) And thus the generalized gamma density becomes

δ − −( x )δ f(x, α, β, δ) = xδα 1e β . (5.2) βδαΓ(α) ·

5.3 Basic Properties for the Generalized Gamma Dis- tribution

We will use the notation X GG(α, β, δ) to denote that a random variable X has a ∼ generalized gamma distribution with density given as in (5.2).

38 If X GG(α, β, δ), k > 0 and m is a positive integer then ∼

kX GG(y, kβ, α, δ), (5.3) ∼

Xm GG(z, βm, α, δ/m). (5.4) ∼ For example (X/β)δ, this random variable will follow a generalized gamma distribution

f(y, 1, α, 1) = yα−1e−y, y > 0, which is the common gamma distribution. If F (x) is the cumulative distribution function for (5.2) given by

x F (x) = f(x, α, β, δ)dx, Z0 then

Γw(α)/Γ(α) if δ > 0 F (x) =   1 Γ (α)/Γ(α) if x < δ − w δ Where w = (x/β) and Γw is the incomplete gamma function. A necessary and suﬃcient condition for W 1/δ to have a G(α, β) distribution is that | | the pdf is of the form |w|δ δα−1 (− ) p (w) = h(w) w e β , W | | with −1 h(w) + h( w) = δ βαΓ(α) , for all w. − | | · ([14] Roberts 1971).

5.4 Particular Cases and Shapes

If δ = 1 in (5.2) we have the well known gamma distribution

1 f(x, α, β) = xα−1 e−x/β, βαΓ(α) ·

39 given by (2.7). If α = 1 (5.2) turns out the Weibull distribution, given by

δ−1 δx −( x )δ h(x, β, δ) = e β . (5.5) βδ We can also get from (5.2) the exponential distribution given by (2.16) just setting δ = 1 in (5.1). This means that an exponential is a special case of a Weibull distribution. Figure ( 5.1) presents a graph for the Weibull when parameters are α = 1, β = 2, δ = 2.

Generalized Gamma Distribution 0.30 0.20 y 0.10 0.00

0 5 10 15 20 25 30

x,alpha=1,beta=5,delta=4

Figure 5.1: Weibull, h(x, 2, 2)

By setting α = 1/2, δ = 2 we get the Half-Normal distribution deﬁned by

2 −( x )2 g(x, β) = e β , x > 0. (5.6) β√π · Where we have used the fact that Γ(1/2) = √π. Often this distribution appears in literature with the patameter β = √2σ, where σ is the standard deviation of a normal random variable. So (5.2) becomes 2 1 2 −( x ) g(x, σ) = e 2σ2 . σ · rπ · Figure ( 5.2) shows a graph for the density of a half normal distribution

40 Generalized Gamma Distribution 0.8 0.6 y 0.4 0.2 0.0

0 1 2 3 4 5

x,alpha=1/2,beta=sqrt(2),delta=2

Figure 5.2: Half Normal, g(x, 1)

Is interesting to analyze the form of the generalized gamma when δ either increases or decreases. The generalized gamma distribution has two shape parameters α and δ. A plot with several graphs when δ is increasing is shown in ﬁgure ( 5.3). Note how the shape is becoming thinner due to the δ factor. When δ ranges between 0 and 1 the shape for the generalized gamma is skewed to the right. The skewness to the right becomes more notorious as δ 0. Figure ( 5.4) shows several graphs analyzing this situation. → When δ decreases the graph of the generalized gamma is very skewed to the right.

41 Generalized Gamma Distribution 1.0 delta=4 0.8 0.6 y delta=3 0.4

delta=2 0.2 delta=1 0.0

0 5 10 15

x,alpha=2,beta=2,delta=varies

Figure 5.3: Analyzing when δ increases

42 Generalized Gamma Distribution 0.30 black.−delta=.1

green.−delta=.7

0.20 blue.−delta=.8 y red.−delta=1 0.10 0.00

0 2 4 6 8 10

x,alpha=2,beta=2,delta=varies

Figure 5.4: Analyzing when δ decreases

43 Chapter 6

Estimation of parameters

6.1 Method of moments

To ﬁnd the moments of the generalized gamma we again use a change of variable for the equation ∞ ∞ δ xrf(x)dx = cxδα+r−1e−(x/β) dx, (6.1) Z0 Z0 and doing as before the change of variable y = (x/β)δ we get

∞ δ − r − − r+δα 1 α+ δ 1 y δα β β y e dy. Γ(α)β δ Z0 So that the rth moment is given by

βrΓ(α + r ) µ0 = δ . (6.2) r Γ(α)

Thus the equations are obtained by equating population and sample moments

βΓ(α + 1 ) x = δ (6.3) Γ(α)

n x2 β2Γ(α + 2 ) i = δ (6.4) n Γ(α) Xi=1 n x3 β3Γ(α + 3 ) i = δ . (6.5) n Γ(α) Xi=1 Stacy ([16] 1965) suggested another method involving the method of moments. If we set Z = ln(X /β)δ = δ(ln(X ) ln(β)), (6.6) i i i −

44 where the Xis, i = 1, . . . n, are independent generalized gamma random variables. And if we denote the central moment of any random variable, say Z by µk(Z), k = 2, 3 . . . . So that (6.6) indicates that k µk(Zi) = δ µk(ln(Xi)). (6.7)

Applying (5.3) and (5.4) to Z = ln(X/β)δ we see that

Z = ln(W ) = exp(αz exp(z))/Γ(α), (6.8) − from where we get that the moment generating function of Z is

E(eθZ ) = Γ(α + θ)/Γ(α). (6.9)

It happens that any kth order derivative of Γ(α + θ), evaluated at θ = 0, is the same whether the derivative is taken with respect to θ or α. Hence the kth moment of Z is

E(Zk) = Γ(k)(α)/Γ(α), (6.10) where Γ(k)(α) is the kth derivative of Γ(α) with respect to α. It follows from (6.6) and (6.7) that δE(ln(X) ln(β)) = µ, (6.11) − 2 0 δ µ2(ln(X)) = µ ,

3 00 δ µ3(ln(X)) = µ .

In terms of the xi, yi = ln(xi), i = 1, 2, . . . n. Solutions for this equations are

a = exp(y µ /δ ) = exp(y s µ /(µ0 )1/2), 0 − 0 0 − y 0 0

0 1/2 δ0 = (µ0) /sy,

g = µ00/(µ0)3/2, −| y| 0 0 where β0 is determined by the last equation , β0 = µ(β0) and µ0 = µ (β0). The sign of δ is 2 according to whether gy is less than or greater than zero respectively. Here y, sy and gy are respectively the sample mean, sample variance and sample skewness for the yi.

45 6.2 Maximum likelihood estimation

The likelihood function for the generalized gamma is given by

n n δ δα−1 − Pn (x /β)δ L(x , . . . x , α, β, δ) = f(x , α, β, δ) = (x , . . . x ) e i=1 i . (6.12) 1 n i Γ(α)n βnαδ 1 n Yi=1 · Taking natural log we have

L(x . . . x , α, β, δ) = n ln(δ) n ln(Γ(α)) nδα ln(β)+(αδ 1) ln(x , . . . x ) Σn (x /β)δ. 1 n − − − 1 n − i=1 i (6.13) Now taking derivative the corresponding equations are ∂L nΓ0(α) n = δn ln(β) + δ ln(x ) = 0. ∂α − Γ(α) − i Xi=1 This can be expressed as ∂L nΓ0(α) n = + δ ln(x ) ln(β) = 0 ∂α − Γ(α) i − Xi=1 (6.14) nΓ0(α) n = + δ ln(x /β). − Γ(α) i Xi=1 Diﬀerentiating with respect to β n n n ∂L δnα − − δ = − xδ( δ)β δ 1 = [ nα + ln(x /β)δ] = nα + ln(x /β)δ = 0. ∂β − β − i − β − i − i Xi=1 Xi=1 Xi=1 And from this last equation β turns out to be ( n (x )δ)1/δ β(α, δ) = i=1 i (6.15) P(nα)1/δ it depends on α and from δ. Diﬀerentiating with respect to δ ∂L n n n x = nα ln(β) + α ln(x ) (x /β)δ) ln( i ) . ∂δ δ − i − i · β Xi=1 Xi=1 Now considering n ln(β) = n ln(β) and using the properties of the function ln we − − i=1 have P n n n x x δ x + α ln( i ) i ln i . (6.16) δ β − β β Xi=1 Xi=1

46 So putting the value of β (6.15) into (6.16) we have

n n n xi xi δ xi + α ln( n δ 1/δ ) n δ 1/δ ln n δ 1/δ . δ (Pi=1(xi) ) − (Pi=1(xi) ) (Pi=1(xi) ) Xi=1 (nα)1/δ Xi=1 (nα)1/δ (nα)1/δ

After rearranging terms we get an expression fot α in terms of δ :

1 α(δ) = . (6.17) δ ( n ln(x )/n) n zδ ln(x ) / n xδ · i=1 i − i=1 i i i=1 i P P P If we substitute (6.17) and (6.15) into (6.14) we will have an equation in terms of δ only. This equation is

n ln(x ) n H(δ) = ψ(α) + δ i=1 i ln xδ + ln(nα), (6.18) − P n − i Xi=1

Γ0(α) where ψ(α) = Γ(α) is the digamma function. And α is given by (6.17). Thus if we are going to try to estimate the parameters for the generalized gamma distribution we need to solve (6.18) for δ. It is not always possible to ﬁnd a solution for H(δ). In many cases it is not even possible to know if there exists a solution. Some authors say that the MLE may not exist unless n > 400. ([1] Continuous Univariate Distributions)

47 Chapter 7

Product and Ratio of two Generalized Gamma

The problem of obtaining an explicit expression, without involving any unsolved integrals, for both the probability density functon and the cumulative distribution of the product of two independent random variables with generalized gamma distributions is a challenging one. Expressions for the pdf of such a product had been obtained by some statisticians in terms of complicated functions. But these functions are not readily computable. They are usually computed in terms of the integrals that deﬁne them. And is even harder try to extend beyond the product of more of two variables. The computations get really complicated. If x δ − − 1 δ1 1 δ1α1 1 ( β ) f(x1) = x e 1 , δ1α1 1 Γ(α1)β1 and x δ − − 2 δ2 2 δ2α2 1 ( β ) f(x2) = x e 2 , δ2α2 2 Γ(α2)β2 are two densities of two independent random variables X1 and X2 with generalized gamma distribution then the joint density is given by

x x δ − − 1 δ1 δ − − 2 δ2 1 δ1α1 1 ( β ) 2 δ2α2 1 ( β ) f(x1, x2) = x e 1 x e 2 . (7.1) δ1α1 1 δ2α2 2 Γ(α1)β1 · Γ(α2)β2

We want to ﬁnd the distribution of Y = X1/X2. If we set W = X1 then we have the variables W W = X , and X = . 1 2 Y

48 The jacobian for these transformations is given by w/y2. So the joint density of Y and W is

(w/y) δ δ − − w δ1 − − δ2 w 1 2 δ1α1 1 ( β ) δ2α2 1 ( β ) fY,W (w, y) = w e 1 (w/y) e 2 = δ1α1 δ2α2 2 Γ(α1)β1 · Γ(α2)β2 · y w δ δ α +δ α −2+1 1 δ α −1+2 −( ) 1 −(w/(yβ ))δ2 c c w 1 1 2 2 ( ) 2 2 e β1 e 2 = (7.2) 1 2 · y · w δ δ α +δ α −1 1 δ α +1 −( ) 1 −(w/(yβ ))δ2 c c w 1 1 2 2 ( ) 2 2 e β1 e 2 . 1 2 · y · Thus ∞ − 1 − w δ1 − δ δ1α1+δ2α2 1 δ2α2+1 ( β ) (w/(yβ2)) 2 fY (y) = c1c2w ( ) e 1 e dw Z0 · y · ∞ 1 − − w δ1 − δ δ2α2+1 δ1α1+δ2α2 1 ( β ) (w/(yβ2)) 2 c1c2( ) w e 1 e dw = (assume δ = δ1 = δ2) y Z0 · ∞ 1 − − w δ − δ δ2α2+1 δ(α1+α2) 1 ( β ) (w/(yβ2)) c1c2( ) w e 1 e dw = y Z0 · ∞ 1 − −wδ 1 + 1 δ2α2+1 δ(α1+α2) 1 (β )δ (yβ )δ c1c2( ) w e 1 2 dw = y Z0 · ∞ δ 1/δ δ 1 δ2α2+1 δ(α1+α2)−1 −w /(β ) 1 1 1 c1c2( ) w e dw = where = + y Z · β (β )δ (yβ )δ 0 1 2 α1+α2 δ(α1+α2) 1 β Γ(α1 + α2) 1 (β1β2y) c1c2 δα +1 = c1c2 δα +1 δ δ α +α Γ(α1 + α2) = y 2 · δ y 2 ((yβ2) + (β1) ) 1 2 δα1+δα2−δα2−1 δ(α1+α2) Γ(α1 + α2) y c1c2 (β1β2) δ δ α +α = · δ ((yβ2) + (β1) ) 1 2 δα1−1 δ(α1+α2) Γ(α1 + α2) y c1c2 (β1β2) δ δ α +α · δ ((yβ2) + (β1) ) 1 2 δα1−1 β1 δ(α1+α2) Γ(α1 + α2) y = c1c2 ( ) . β1 · β2 · δ (yδ + ( )δ)α1+α2 β2 We will say that a random variable which has this distribution has a GGR (generalized gamma ratio) with parameters (δ, k, α1, α2) where k = β1/β2. Coelho and Mexia ([7] Coelho and Joao 2007) gave explicit and concise expressions for the pdf and cumulative distributions of

W = Xj Yj=1

49 where the Xj are independent random variables with all distinct shape parameters. If X GGR(k , α , α , δ ), j = 1 . . . m. Where all the parameters are positive. Let j ∼ j 1j 2j j

s1j = αij and s2j = α2j, then if δ s = δ s and δ s = δ s , for all j = k, j, k = 1, 2 . . . n, the density and j 1j 6 k 1k j 2j 6 k 2k 6 cumulative distribution of the random variable m

W = Xj, (7.3) Yj=1 are given by

m n ∗ s∗ 1 ∗ →∞ 1jh limn K1K2 j=1 h=0 H2jhdhj(w/K ) w if 0 < w K fW (w) =  ≤ Pm Pn ∗ −s∗ 1 ∗  lim →∞ K K H c (w/K ) 2jh if w K , n 1 2 j=1 h=0 1jh hj w ≥ P P And 

d ∗ ∗ ∗ m n hj s1jh limn→∞ K1K2 H2jh ∗ (w/K ) if 0 < w K j=1 h=0 s1jh FW (w) =  ≤ , Pm Pn dhj chj w −s ∗ →∞ 2jh  limn K1K2 j=1 h=0 H2jh s∗ + H1jh s∗ 1 ( K∗ ) if w K , 1jh 2jh − ≥ P P where  m ∗ −1/δj K = kj , Yj=1 m n m n ∗ ∗ K1 = s1jh, K2 = s2jh, Yj=1 hY=0 Yj=1 hY=0 and, for j = 1. . . . , m and n = 0, 1, . . . , m n 1 m n 1 c = , c = , hj s∗ s∗ hj s∗ s∗ ηY=1 Yν= 2ην − 2jh ηY=1 Yν= 1ην − 1jh and m n d m n c H = kl , H = kl , 1jh s∗ + s∗ 2jh s∗ + s∗ Xk=1 Xl=0 2jh 1kl Xk=1 Xl=0 1jh 2kl with ∗ ∗ ∗ ∗ s1jh = δj (s1j + h) and s2jh = δj (s2j + h), (j = 1, . . . m).

There are other considerations when the parameters are not the same. (See [7] Coelho and Joao 2007).

50 Chapter 8

Hierarchical Models

All of the cases seen thus far a random variable had a single distribution depending on parameters. Sometimes is more convenient to think of the things in a hierarchy. A well known hierarchical model is the following. An insect lays a large number of eggs , each surviving with probability p. On the average, how many eggs will survive? The ”large number” of eggs laid is a random variable , often chosen to be Poisson (λ). Furthermore we assume that each eggs survival is independent, then we have Bernoulli trials. So if we set X = number of survivors and Y = number of Bernoulli trials we have

X Y binomial(Y, p) | ∼

Y Poisson(λ). ∼ We want to compute

∞ P (X = x) = P (X = x, Y = y) Xy=0 ∞ = P (X = x Y = y)P (Y = y) | Xy=0 ∞ e−λλy = [(yx)px(1 p)y−x][ ]. − y! Xy=0

Since X Y = y is binomial(y, p) and Y is Poisson(λ). If we now simplify this last expression |

51 we have ∞ (λpe−λ) ((1 p)λ)y−x P (X = x) = − x! (y x)! Xy=x − ∞ (λpe−λ) ((1 p)λ)t = − x! (t)! Xy=x (λpe−λ) = e(1−p)λ x! (λpe−λ) = e−λp, x! so X Poisson(λp) And from this is easily seen that ∼ EX = λp.

Now if X is a random variable with Generalized Gamma density in the form of (5.2) we will use the notation X GG(α, β, δ). We will also write c(α, β, δ) = δ to refer ∼ βδαΓ(α) to the constant in the generalized gamma density. Let’s consider Y Λ Poisson(Λ) and | ∼ Λ Gg(α, β, δ). Then the distribution of Y is given by ∼ ∞ y −y λ e α−1 −( λ )δ f(y) = c(α, β, δ) λ e β dλ. Z0 y! · · ·

−x ∞ (−1)k k To evaluate this integral we wiil use the fact that e = k=0 k! x . So for this case we have P ∞ ∞ y −λ k λ e − ( 1) λ δk f(y) = c(α, β, δ) λα 1 − dλ Z y! · · · k! β 0 Xk=0 ∞ ∞ k c(α, β, δ) − − ( 1) λ δk = e λλy+α 1 − dλ y! Z · k! β 0 Xk=0 ∞ ∞ k c(α, β, δ) ( 1) λ δk − − = − λy+α 1 e λdλ y! Z k! β · · Xk=0 0 ∞ k ∞ c(α, β, δ) ( 1) − − = − λδk+y+α 1 e λdλ. y! k!βk Z · Xk=0 0 Seeing the integral in the right hand side as Γ(δk + y + α) we get ∞ c(α, β, δ) ( 1)k f(y) = − Γ(δk + y + α). (8.1) y! k!βk · Xk=0

52 So we can get the distribution of Y just by changing parameters. For example if δ = β = 1 we have that Λ is a random variable with standard gamma density, so ∞ ∞ c(α, 1, 1) ( 1)k 1 ( 1)k f(y) = − Γ(k + y + α) = − Γ(k + y + α). y! · k! Γ(α)y! · k! Xk=0 Xk=0 If α = δ = 1 we have an exponential distribution. So (8.1) becomes ∞ 1 ( 1)k f(y) = − Γ(k + y + 1) βy! · βkk! Xk=0 ∞ k ∞ 1 ( 1) − = − λk+ye λdλ βy! · βkk! Z Xk=0 0 ∞ ∞ k 1 ( 1) − = − λkλy e λdλ βy! Z βkk! · · 0 Xk=0 ∞ ∞ k 1 ( 1) λ k − = − λy e λdλ βy! Z k! β · · 0 Xk=0 ∞ ∞ 1 λ −λ y 1 − 1+ 1 λ y = e β e λ dλ = e β λ dλ. βy! Z0 · · βy! Z0 · 1 Now making the change of variable u = (1 + β )λ we arrive to 1 ∞ uy β f(y) = e−u du β y! Z · (1 + 1 )y · β + 1 · 0 β 1 ∞ = e−uuydu (1 + 1 )y (β + 1) y! Z β · · 0 Γ(y + 1) 1 = = . (1 + 1 )y (β + 1) y! (1 + 1 )y (β + 1) β · · β · Remember that for any two random variables we can relate them with the following formula E(X) = E(E(X Y )), (8.2) | and V ar(X) = E(V ar(X Y )) + V ar(E(X Y )), (8.3) | | provided that the expectations exist. For the same case from above we have that βΓ(α + 1 ) E(Y ) = E(E(Y Λ)) = E(Λ) = δ , | Γ(α)

53 which is just the expected value given by (6.3). The variance for Y is given by

V ar(Y ) = E(V ar(Y Λ)) + V ar(E(Y Λ)) | | = E(Λ) + V ar(Λ) = E(Λ) + E(Λ2) (E(Λ))2 − βΓ(α + 1 ) β2Γ(α + 2 ) β2Γ(α + 1 )2 = δ + δ + δ . Γ(α) Γ(α) Γ(α)2

54 Chapter 9

Random Numbers Generation for a Generalized Gamma Distribution

If we set the change of variable y = (x/β)δ in (5.2) a standard gamma distribution with parameter α is obtained.

δ δα−1 −( x )δ f(x, α, β, δ)dx = x e β βδαΓ(α) · 1−δ 1−δ 1/δ δ − −( βy )δ βy δ βy δ = (βy1/δ)δα 1e β ( because dx = ) βδαΓ(α) · · δ δ 1 = yα−1/δ+1/δ−1 e−y Γ(α) · · 1 = f(y) = yα−1e−y. Γ(α) ·

So we can generate random numbers for y (which is easy with the adequate software) in this gamma density and then relate this numbers to the generalized gamma by means of the equation x = βy1/δ. (9.1)

This can work to estimate the cummulative distribution when X GG(α, β, δ). That is, ∼ for any t > 0 t δ − − x δ δα 1 ( β ) P (X t) = δα x e dx. (9.2) ≤ Z0 β Γ(α) ·

If we generate n random numbers x1, x2, . . . xn from X, t > 0 then the empirical distribution is deﬁned as n F (x) = 1 x t . (9.3) n { i ≤ } Xi=1 b

55 Here 1 x is the indicator function which is equal to 1 if x t and 0 otherwise. Consider { } ≤ for example X GG(2, 4, 2). Generating n = 5000 for a standard gamma with parameter ∼ α = 2 and using (9.1) we obtain the next table

Table 9.1: Empirical distribtuion for a generalized gamma random variable. α = 2, β = 4, δ = 2, E(X) = 5.31, V ar(X) = 3.725

t Empirical distribution 1 .0024 2 .0264 3 .1070 4 .2648 5 .4644 6 .6586 7 .8054 8 .9030 9 .9592 10 .9840

56 Emp. Dist. of a Generalized Gamma 1.0

alpha=2,beta=4,delta=2 0.8 0.6 emp_dist 0.4 0.2 0.0

2 4 6 8 10

Figure 9.1: Empirical distribution for a Generalized Gamma

9.1 Estimation of the sum of n Generalized Gamma Distributions

As we have (9.1) to obtain the empirical distribution for a generalized gamma, we can obtain empirical distribution for Y = X + X + . . . + X , where X GG(α , β , δ ) for 1 2 n i ∼ i i i i = 1, 2, . . . n. And we can use the normal approximation we used in (4.13) and see how it works here for a sum of generalized gamma distributions.

57 Table 9.2: Normal approximation for the sum of the exact distribution of 2 generalized gamma variables. α1 = 2, α2 = 5, β1 = 4, β2 = 3, δ1 = 2, δ2 = 3, E(Y ) = 10.33, V ar(Y ) = 4.3

t Empirical Normal Approximation Diﬀerence 5 .0011 .0013 -.0002 7 .0436 .0429 .0005 9 .2736 .2708 .0028 11 .6446 .6442 .0004 13 .8927 .8957 -.0030 15 .9810 .9809 .0001 17 .9981 .9977 .0004 19 .9998 .9998 .0000

Emp. Dist. of a Generalized Gamma 1.0 red=empirical 0.8

0.6 blue=approximate emp_dist 0.4 0.2 0.0

5 10 15

58 Table 9.3: Normal approximation for the sum of the exact distribution of 2 generalized gamma variables. α1 = 2, α2 = 5, β1 = 4, β2 = 3, δ1 = .7, δ2 = .8, E(Y ) = 35.40, V ar(Y ) = 326.92

t Empirical Normal Approximation Diﬀerence 11 .0305 .0265 .0040 21 .2162 .2113 .0049 31 .4763 .4766 -.0003 41 .6914 .6933 -.0018 51 .8310 .8324 -.0014 61 .9120 .9119 .0001 71 .9537 .9768 -.0008 81 .9764 .9768 -.0004 91 .9883 .9882 .0002

Emp. Dist. of a Generalized Gamma 1.0 red=empirical 0.8 0.6 emp_dist 0.4 blue=approximate 0.2 0.0

0 20 40 60 80

59 Table 9.4: Normal approximation for the sum of the exact distribution of 5 generalized gamma variables. α1 = 2, α2 = 5, α3 = 10, α4 = 3, α5 = 78, β1 = 4, β2 = 3, β3 = 81, β4 = 37, β5 = 125, δ1 = 2, δ2 = 3, δ3 = 4, δ4 = 5, δ5 = 6, E(Y ) = 456.012, V ar(Y ) = 188.68

t Empirical Normal Approximation Diﬀerence 410 .0006 .0006 .0000 420 .0054 .0051 .0003 430 .0313 .0308 .0006 440 .1233 .1224 .0009 450 .3268 .3279 -.0010 460 .6087 .6109 -.0021 470 .8446 .8459 -.0013 480 .9622 .9614 .0009 490 .9942 .9942 .0000

Emp. Dist. of a Generalized Gamma 1.0 red=empirical 0.8 0.6 blue=approximate emp_dist 0.4 0.2 0.0

400 420 440 460 480 500

60 Table 9.5: Normal approximation for the sum of the exact distribution of 10 generalized gamma variables. α1 = 2, α2 = 5, α3 = 10, α4 = 3, α5 = 7, α6 = .20, α7 = .9, α8 = 2.5, α9 = .3, α10 = .05β1 = 4, β2 = 3, β3 = 5, β4 = 6, β5 = 2, β6 = 500, β7 = 18β8 = 25, β9 = 20, β10 = 100.5, δ1 = 2, δ2 = 3, δ3 = 4, δ4 = 5, δ5 = 6, δ7 = 8, δ8 = 9, δ9 = 10, δ10 = 11, E(Y ) = 405.1474, V ar(Y ) = 19932.65.

t Empirical Normal Approximation Diﬀerence 300 .2572 .2275 .0298 400 .4798 .4840 -.0042 500 .7174 .7487 -.0312 600 .9127 .9168 -.0040 700 .9915 .9821 .0094 800 .9998 .9976 .0022 900 1 .9998 .0002 1000 1 1 .0000

Emp. Dist. of a Generalized Gamma 1.0

0.8 red=empirical

0.6 blue=approximate emp_dist 0.4

300 400 500 600 700 800 900

61 References

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[8] John E. Freunds, Mathematical Statistics with Applications, Seventh Edition, Pearson Prentice Hall (2004), Upper Saddle River, New Jersey 221-255.

[9] P.G. Moschopoulos, (1985) The Distribution of the sum of Independent Gamma Ran- dom Variables, Annals of the Institute of Statistical Mathematics, 37, Part A, 541-544

62 [10] P.G. Moschopoulos, W.B. Canada (1984) The distribution function of a linear combination of Chi-squares, Comps and Maths with applications, Vol 10. No. 4/5 pp. 383-386

[11] P.G. Moschopoulos,(1983) On a new Transformation to Normality, Communication Statistics-Theory and Methods, 12(16), pp. 1873-1878

[12] P.G. Moschopoulos, Govind, S. Mudholkar, (1983) A Likelihood Ratio Based Normal Approximation for the Non-Null Distribution of the Multiple Correlation Coeﬃcient, Communication Statistics-Simulations and Computations, 12(3), pp 355-371

[13] Pham-Gia, Turkkan, N. (2002). Operations on the generalized-F variables and applications. Statistics, 36, 195-209.

[14] Roberts,C.D. (1971). On the distribution of random variables whose mth absolute power is gamma, Sankhya, Series A, 33, pp. 229-232

[15] E.W. Stacy, (1962) A generalization of the gamma distribution. The Annals of Math- ematical Statistics, Vol. 33, No. 3, pp. 1187-1192

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[17] Stacy E. W. Mihram G.A. (1965). Parameter estimation for a generalized gamma distribution, Technometrics, 7, pp. 349-358.

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63 Chapter 10

Appendix

All the codes have been written in the software R.

10.1 Code to Approximate the sum of k Independent Gamma Random Variables

#Number of terms n=10

#Number of generation for the random numbers n2=1000

#Setting the parameters alpha=c(1:n) beta=c(1:n) alpha[1]=2 alpha[2]=2.5 alpha[3]=3 alpha[4]=3.5 alpha[5]=4 alpha[6]=4.5

64 alpha[7]=5 alpha[8]=5.5 alpha[9]=6 alpha[10]=6.5

beta[1]=2 beta[2]=2.5 beta[3]=3 beta[4]=3.5 beta[5]=4 beta[6]=4.5 beta[7]=5 beta[8]=5.5 beta[9]=6 beta[10]=6.5

#Normal approx k1=sum(alpha*beta) k2=sum(alpha*beta^2) k3=sum(alpha*beta^3) phi_2=k2/k1 phi_3=k3/k1 phi_4=k4/k1

65 h=1-k1*k3/(3*k2^2) mu_h=1+h*(h-1)*phi_2/(2*k1)+h*(h-1)*(h-2)*(4*phi_3+3*(h-3)*phi_2^2)/(24*k1^2) sigma_h=phi_2*h^2/(k1)+(h-1)*h^(2)*(2*phi_3+(3*h-5)*phi_2^2)/(2*k1^2)

#Estimation with random numbers Gen=matrix(0,n,n2) for(i in 1:n) { Gen[i,]=rgamma(n2,alpha[i],1/beta[i]) }

#Estimation of alpha alpha_est=c(1:n) s_est=c(1:n) for(i in 1:n) { s_est[i]=log(mean(Gen[i,])) -sum(log(Gen[i,]))/n2 alpha_est[i]=(3-s_est[i]+sqrt((s_est[i]-3)^2 +24*s_est[i]))/(12*s_est[i]) }

#Estimation of the parameter "beta" beta_est=c(1:n) for(i in 1:n) { beta_est[i]=mean(Gen[i,])/(alpha_est[i])

66 } beta_est alpha_est

#Exact distribution #Number of terms we will use in the infinite sum k=100

#Length of the subintervals the interval is divided in c=.1 #Right extreme of the interval lim_r=300

#Setting the minimum between beta’s beta_min=min(beta)

#Setting the coefficient rho rho=sum(alpha)

#Coefficients C and Lower gamma beta_min_ov_beta=(beta_min^{rho})/(prod(beta^{alpha})) C=prod(beta_min_ov_beta) G=1-beta_min/beta lower_gamma_matrix=matrix(1,n,(k+1)) for(j in 1:n)

67 { for(i in 1:(k+1)) { lower_gamma_matrix[j,i]=(alpha[j]*(G[j])^{i})/i } } lower_gamma=colSums(lower_gamma_matrix)

#Setting the delta’s delta=c(1:(k+1)) delta[1]=lower_gamma[1] for(j in 2:(k+1)) { b=c(1:(j-1)) for(v in 1:(j-1)){b[v]=v*lower_gamma[v]*delta[(j-v)]} d=sum(b)+j*lower_gamma[j] delta[j]=(1/j)*(d) }

#Domain y=seq(0,lim_r,by=c) #Density density_sum_gamma=matrix(1,(k+1),(1/(c/lim_r))+1) for(u in 1:(k+1)) { density_sum_gamma[u,]=((gamma(rho+u)*(beta_min)^{rho+u})^{-1}) *delta[u]*y^{rho+u-1}*exp(-y/beta_min) }

68 E=C*colSums(density_sum_gamma)+C*y^{rho-1}*exp(-y/beta_min) *((gamma(rho)*(beta_min)^{rho})^{-1})

#Approximate expect_z=sum(alpha*beta) var_z=sum(alpha*beta^{2}) beta_z=var_z/expect_z alpha_z=expect_z^(2)/var_z App=1/(beta_z^{alpha_z}*gamma(alpha_z))*y^{alpha_z-1}*exp(-y/beta_z)

#Graphs for the distribution #Exact pgamma_vector_ex=c(1:lim_r) for(j in 1:lim_r) { A=c(1:(k+1)) for(i in 1:(k+1)) { A[i]=delta[i]*pgamma(j,rho+i,1/beta_min) } distribution_1=c(A,pgamma(j,rho,1/beta_min)) pgamma_vector_ex[j]=C*sum(distribution_1) } domain=c(1:lim_r) Ex=pgamma_vector_ex

69 #Approximate pgamma_vector_app=c(1:lim_r) for(j in 1:lim_r) { pgamma_vector_app[j]=pgamma(j,alpha_z,1/beta_z,) } App=pgamma_vector_app

#Estimated pgamma_vector_est=c(1:lim_r) for(j in 1:lim_r) { pgamma_vector_est[j]=pgamma(j,alpha_z_est,1/beta_z_est) } Est=pgamma_vector_est

#Estimated_h k1=sum(alpha*beta) k2=sum(alpha*beta^2) k3=2*sum(alpha*beta^3) phi_2=k2/k1 phi_3=k3/k1 phi_4=k4/k1

h=1-k1*k3/(3*k2^2)

70 #h=1/3 mu_h=1+h*(h-1)*phi_2/(2*k1)+h*(h-1)*(h-2)*(4*phi_3+3*(h-3)*phi_2^2)/(24*k1^2) sigma_h=phi_2*h^2/(k1)+(h-1)*h^(2)*(2*phi_3+(3*h-5)*phi_2^2)/(2*k1^2) pgamma_vector_est_h=c(1:lim_r) for(j in 1:lim_r) { z_h=((j/k1)^{h} -mu_h)/(sqrt(sigma_h)) pgamma_vector_est_h[j]=pnorm(z_h) } Est_h=pgamma_vector_est_h

par(mfrow=c(1,2)) plot(domain,Ex,type="l",col="red",xlab="x",main="Exact distribution") #plot(domain,App,type="l",col="blue",xlab="x",main="Two moments app") plot(domain,Est_h,type="l",col="brown",xlab="x",main="Normal Approx") #plot(domain,Ex_est,type="l",col="black") #plot(domain,Est,type="l",col="green")

#Distribution several times #Approximate a_1=180 b_1=340 c_1=20 vector_t=seq(a_1,b_1,by=c_1) Approx_1=c(1:((b_1-a_1)/c_1 +1)) for(i in 1:((b_1-a_1)/c_1+(1))) {

71 Approx_1[i]=pgamma(vector_t[i]/beta_z,alpha_z) } #Exact_dist=C*sum(distribution)

#Exact Exact_1=c(1:((b_1-a_1)/c_1 +1)) for(j in 1:((b_1-a_1)/c_1+(1))) { A=c(1:(k+1)) for(i in 1:(k+1)) { A[i]=delta[i]*pgamma(vector_t[j],rho+i,1/beta_min,) } distribution_2=c(A,pgamma(vector_t[j],rho,1/beta_min)) Exact_1[j]=C*sum(distribution_2) }

#Estimated exact Exact_1_est=c(1:((b_1-a_1)/c_1 +1)) for(j in 1:((b_1-a_1)/c_1+(1))) { A_est=c(1:(k+1)) for(i in 1:(k+1)) { A_est[i]=delta_est[i]*pgamma(vector_t[j],rho_est+i,1/beta_min_est) } distribution_2_est=c(A_est,pgamma(vector_t[j],rho_est,1/beta_min_est,))

72 Exact_1_est[j]=C_est*sum(distribution_2_est) }

#Estimated with two moments Estimated_1=c(1:((b_1-a_1)/c_1 +1)) for(i in 1:((b_1-a_1)/c_1+(1))) { Estimated_1[i]=pgamma(vector_t[i],alpha_z_est,1/beta_z_est) }

#Estimated with h Normal_App_1=c(1:((b_1-a_1)/c_1 +1)) z_h=c(1:((b_1-a_1)/c_1 +1)) for(i in 1:((b_1-a_1)/c_1+(1))) { z_h[i]=((vector_t[i]/k1)^{h} -mu_h)/(sqrt(sigma_h)) Normal_App_1[i]=pnorm(z_h[i]) } Approx=round(Approx_1,digits=7) Exact=round(Exact_1,digits=7) Estimated=round(Estimated_1,digits=7) Est_ex=round(Exact_1_est,digits=7) Normal_App=round(Normal_App_1,digits=7) Diff_1=Exact_1-Approx_1 Diff_E_A=round(Diff_1,digits=7) Diff_10_3_1=Diff*1000 Diff_10_3=round(Diff_10_3_1) Diff_2=Exact-Estimated

73 Diff_E_Est=round(Diff_2,digits=7) Diff_3=Exact-Est_ex Diff_E_EE=round(Diff_3,digits=7) Diff_4=Exact_1-Normal_App_1 Diff_E_Normal=round(Diff_4,digits=7)

#Printing out results #Table_results=cbind(vector_t,Exact,Approx,Est_ex,Estimated,Normal_App, Diff_E_A,Diff_E_EE,Diff_E_Est,Diff_E_Normal) #Table_results=cbind(vector_t,Exact,Approx,Diff_E_A) #Table_results=cbind(vector_t,Exact,Est_ex,Diff_E_EE) Table_results=cbind(vector_t,Exact,Normal_App,Diff_E_Normal) #Table_results=cbind(vector_t,Exact_1,Approx_1, Estimated_h_1,Diff_1,Diff_4) Table=data.frame(Table_results) #write.table(Table,"C:/Users/Hugo/Desktop/Hugo/Thesis/ Simulation_Independent_Sum_of_Gamma/ex_norm_tabla_1.txt",sep=",")

74 10.2 Code to Approximate the Sum of k Independent Generalized Gamma Random Variables

#Sum of Generalized Gamma #Empirical distribution #Number of terms n=5

#Number of random data generated n1=50000

#Standard gamma numbers y_ran_data=matrix(0,n,n1)

#Parameters alpha=c(1:n) beta=c(1:n) delta=c(1:n) alpha[1]=2 alpha[2]=5 alpha[3]=10 alpha[4]=3 alpha[5]=78 #alpha[6]=.2 #alpha[7]=.9 #alpha[8]=2.5 #alpha[9]=.3

75 #alpha[10]=.05 beta[1]=4 beta[2]=3 beta[3]=81 beta[4]=37 beta[5]=125 #beta[6]=500 #beta[7]=18 #beta[8]=25 #beta[9]=20 #beta[10]=100

#m=2 delta[1]=.5 delta[2]=.6 delta[3]=.3 delta[4]=.8 delta[5]=.2 #delta[6]=m+5 #delta[7]=m+6 #delta[8]=m+7 #delta[9]=m+8 #delta[10]=m+9

#Gen gamma numbers x_vectors=matrix(0,n,n1) for(i in 1:n)

76 { y_ran_data[i,]=rgamma(n1,alpha[i]) x_vectors[i,]=beta[i]*(y_ran_data[i,]^{1/delta[i]}) } sum_gen_gamma=colSums(x_vectors) a_1=300 b_1=1000 c_1=100 vector_t=seq(a_1,b_1,by=c_1) emp_dist=c(1:((b_1-a_1)/c_1 +1)) for(i in 1:((b_1-a_1)/c_1 +1)) { cont=0 for(j in 1:n1) { if(sum_gen_gamma[j]<=vector_t[i]){cont=cont+1} } emp_dist[i]=cont/n1 }

#Cumulants mean=sum(beta*(gamma(alpha+(1/delta)))/gamma(alpha)) var=sum(beta^{2}*(gamma(alpha+(2/delta)))/gamma(alpha)- (beta*(gamma(alpha+(1/delta)))/gamma(alpha))^2) k1=mean k2=var

77 k3=sum(beta^{3}*gamma(alpha+(3/delta))/(gamma(alpha))-3*beta^{2} *(gamma(alpha+(2/delta)))/gamma(alpha) *(beta*(gamma(alpha+(1/delta)))/gamma(alpha))+2*(beta*(gamma(alpha+(1/delta))) /gamma(alpha))^(3)) phi_2=k2/k1 phi_3=k3/k1 phi_4=k4/k1

h=1-k1*k3/(3*k2^2) mu_h=1+h*(h-1)*phi_2/(2*k1)+h*(h-1)*(h-2)*(4*phi_3+3*(h-3)*phi_2^2)/(24*k1^2) sigma_h=phi_2*h^2/(k1)+(h-1)*h^(2)*(2*phi_3+(3*h-5)*phi_2^2)/(2*k1^2) Normal_App_sum_gg=c(1:((b_1-a_1)/c_1 +1)) z_h=c(1:((b_1-a_1)/c_1 +1)) for(i in 1:((b_1-a_1)/c_1+(1))) { z_h[i]=((vector_t[i]/k1)^{h} -mu_h)/(sqrt(sigma_h)) Normal_App_sum_gg[i]=pnorm(z_h[i]) } diff=emp_dist-Normal_App_sum_gg results=cbind(vector_t,emp_dist,round(Normal_App_sum_gg,4),round(diff,4)) #results=cbind(vector_t,emp_dist) Results=data.frame(results) #write.table(Results,"C:/Users/Hugo/Desktop/Hugo/Thesis /Simulation_Independent_Sum_of_Gamma/sum_gen_gamm_7.txt",sep=",")

78 Results plot(vector_t,emp_dist,type="l",col="red",xlab="t",main ="Emp. Dist. of a Generalized Gamma") lines(vector_t,Normal_App_sum_gg,col="blue") text(locator(1),"red=empirical") text(locator(1),"blue=approximate")

79 10.3 Bedfordshire Rainfall Statistics Table

Source:http://www.zen40267.zen.co.uk/rainfall/rainfall.html. Table made by Andrew Leaper

Table 10.1: Rainfall Statistics in mm

Month 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 January 46.3 36.8 70.6 668.5 19.6 19 69.4 32.8 37.3 57.9 26.3 67.1 February 82 60.1 22.6 66.8 28.4 43.5 30.5 19.3 71.2 69.7 41.3 16 March 33 22.2 83.8 32.7 40.7 55.1 33.4 19.4 37.4 81.6 43.7 38.1 April 15.9 30.1 34.3 3.7 43.4 30.1 77.9 19.4 24.8 93.7 123 43.1 May 51.1 31.2 110 146 84.5 22.1 48.7 43.1 62.4 58.7 83.8 49.6 June 43.7 55.4 70.7 136 8.9 57.6 29.1 43.7 33.4 22.8 24.9 86.3 July 22.9 121 77.6 110 57.5 43 63.2 67.6 80 89.4 55.8 18.1 August 173 65.7 91.4 31.5 73.2 42.8 125 0 75.9 58.7 55.6 64.9 September 58.5 9.3 48.6 34.3 62.9 66.3 12.7 22.7 21.3 67.3 48.2 56.9 October 44.6 42.4 56.6 93 101 55.9 109 36.5 102 93.4 153 53.1 November 36.5 95.8 80.6 50.3 90.9 37.8 55.8 88.1 102 28.6 103 14.1 December 22.2 78.5 23.9 32.9 58.1 29.7 21.4 59 128 39.4 98.1 82.9

80 Table 10.2: Rainfall Statistics in mm (continuation)

Month 1998 1997 1996 1995 1994 1993 1992 1991 1990 1989 1987 January 56.8 15.7 39.6 86.9 67.8 70.8 62.1 51.3 41.9 37.7 26.3 February 4.3 27.8 55.5 69.8 38.3 5.8 13.2 15.6 85.4 36.6 90.3 March 53.5 20.4 26.5 53.1 32.5 15 33 61.4 18 18 60.9 April 143.5 15 28.4 27.8 61.7 89.6 46.9 24.3 31.8 111.7 123 May 28.1 112.7 17.9 29.4 79.2 54.4 115.2 37 1 16.6 83.8 June 137.1 99 17 25.8 23.5 76.3 48.4 94.5 30.6 26.8 24.9 July 8.6 21.2 17 47 16.2 52.9 96.5 37.4 29.2 76.7 55.8 August 33.1 59.9 43.2 2.7 33.1 50.7 107.1 53.4 26.3 32.8 55.6 September 104.3 15.9 52.4 106.6 109.1 68 143.7 102.5 22.9 35.1 48.2 October 63.5 62.1 18.6 33.2 54.8 122.4 74.9 15.6 55.3 36.5 153 November 109.9 63.5 47.8 41.8 45.2 46.65 143.3 66.6 34.2 46.2 103 December 55.3 72.1 39.4 104.2 72.7 99.5 54.7 15.7 54.8 120.3 98.1

81 Table 10.3: Rainfall Statistics in mm (continuation)

Month 1988 1987 1986 1985 1984 January 97.7 32.5 54.5 42 51.3 February 25.7 17 35 26.8 38.5 March 56.8 60.55 43.1 28.9 39.5 April 15.8 56 75.5 22.5 20.5 May 65.2 47.7 71 50.3 57.5 June 42.1 132 22.5 98 69 July 145 52 41.5 24.2 22 August 13.8 89 85.5 65 46.5 September 71.5 33 25.5 15.5 79 October 59.3 155 43 24 37.5 November 13.8 69.4 79.5 39.5 87.5 December 42.8 13.66 30.5 101 43.5

82 Curriculum Vitae

V´ıctor Hugo Jiménez Nava was born on May 8, 1984. The second son of Jorge Sabino Jiménez and Mar´ıa Elva Nava Morales, he graduated from Juarez Autonomous University, Juárez, Chihuahua México, in the spring of 2009. He obtained a bachelor in mathematics. He has attended research summer programs in Mexico. He went to Colima, Colima at the Colima Autonomous University in 2007 and Guanajuato, Guanajuato, at the Research Center in Mathematics in 2008. He entered University of Texas at El Paso in the fall of 2009. While pursuing a master’s degree in statistics he worked as a Teaching Assistant, and as instructor for two mathematical courses directed to freshman students. He submitted this work on december 2011 as a part of the requirements to complete his master in statistics at the University of Texas at El Paso.